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Exam2_Solutions_W11

# Exam2_Solutions_W11 - Math 115 Second Midterm Name EXAM...

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Math 115 / Exam 2 (March 22, 2011) page 2 w 1 . [15 points] A hoophouse is an unheated greenhouse used to grow certain types of vegetables during the harsh Michigan winter. A typical hoophouse has a semi-cylindrical roof with a semi-circular wall on each end (see figure to the right). The growing area of the hoophouse is the rectangle of length and width w (each measured in feet) which is covered by the hoophouse. The cost of the semi-circular walls is \$0.50 per square foot and the cost of the roof, which varies with the side length , is \$1 + 0 . 001 per square foot. a . [4 points] Write an equation for the cost of a hoophouse in terms of and w . (Hint: The surface area of a cylinder of height and radius r , not including the circles on each end, is A = 2 πrℓ .) Solution: The roof has area πrℓ = π 2 wℓ . The walls have area πr 2 = π 4 w 2 . This means the cost is C = 0 . 50 · π 4 w 2 + (1 + 0 . 001 ) π 2 wℓ = π 8 w 2 + π 2 (1 + 0 . 001 ) wℓ. b . [11 points] Find the dimensions of the least expensive hoophouse with 8000 square feet of growing area.
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